Some properties of composition operators on Hilbert spaces of Dirichlet series

Abstract

In this paper, we study some properties of composition operators on Hilbert spaces of Dirichlet series, which include the Fredholmness, Hilbert-Schmidtness, spectra, cyclic and hypercyclic phenomenons, and also answer a norm question raised by Cowen and MacCluer.

Keywords:

• Hilbert space of Dirichlet series
• composition operator
• spectrum
• cyclicity
• hypercyclicity

AMS Subject Classifications:

• Primary 47B33
• Secondary 30D55
• 46E15

Acknowledgements

The authors would like to thank Professor Ruhan Zhao for his invaluable conversation. The authors also wish to thank Dr Xiaolu Hou for providing us the reference [16].

Notes

This research is partially supported by National Science Foundation of China (NSFC) [grant number 11271293].